Role of charge accumulation in guided streamer evolution in helium DBD plasma jets

Experimental data are presented on the evolution of a helium atmospheric pressure plasma jet driven by a tailored voltage waveform generated as bunches of voltage pulses consisting of a superposition of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\approx 43$$\end{document}≈43 kHz bipolar square pulses and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\approx 300$$\end{document}≈300 kHz oscillations. The characteristics of directed ionization waves (guided streamers) are compared for bunches with different first pulse polarities and different bunch duty cycles. The longest and brightest streamers are achieved at the voltage bunch with the first negative pulse and a minimum duty cycle. The dynamics of streamers at the voltage bunch with the first positive pulse are characterized by the shortest length and a lower brightness. The plasma jet length can be smoothly changed by varying the number of pulses in the bunch and the polarity of the first pulse. It is thus possible to precisely localize the region of a strong field in space by combining the parameters of the applied voltage (the duty cycle and polarity of the first pulse of a bunch) with a stepwise propagation mode of a guided streamer.

This Supplementary Information contains an estimation of the charged cloud expansion consisting of cold ions. Consider the expansion of a spherically symmetric charged homogeneous cloud of cold ions with initial radius R 0 in a gas as a result of mutual charge-charge repulsion (mutual Coulomb repulsion). Let us assume that the uniformity of the sphere is preserved during expansion relative to the sphere center. Additionally, assume that the Debye radius is close to the radius of the cloud. Then, the attraction of the ions can not keep the electrons inside the cloud 1 , and the electrons will leave the cloud when the time ∼ R 0 /v e < 10 ns. The resulting ion cloud without electrons will expand because of the force of Coulomb repulsion.
In the case of a low degree of ionization, charged particles can be considered as an independent component of the gas, and it can be assumed that charged particles of each type diffuse through a neutral gas without noticeable interaction between each other or charged particles of other types 2 . The Coulomb repulsion in addition to diffusion must be considered 2 starting at an ion concentration of 10 5 cm −3 . Coulomb mutual repulsion prevails over diffusion at an ion concentration above 10 6 cm −3 . At a concentration above ∼ 10 9 cm −3 , the Debye radius R D = 4.86(T e [K]/n e [cm −3 ]) 1/2 [cm] size 3 becomes much smaller than the system, and this model does not describe the system well 3 .
In the vicinity of streamer discharges in air, the concentration of uncompensated charges 3-5 is estimated to be at a level of 10 8 cm −3 . For a helium plasma jet flowing into the air, an uncompensated charge 6 on the order of this value forms near the discharge tube exit. Thus, it is reasonable to consider the expansion of a cloud with charges Q of 0.001, 0.01, 0.1 and 1 nC and a radius of R 0 = 0.5 cm. These charges Q of 0.001, 0.01, 0.1 and 1 nC correspond to concentrations n + of ≈ 10 7 , 10 8 , 10 9 and 10 10 cm −3 , respectively.
The charged particle drift velocity at cloud radius v + = µ + E is equal to the cloud border expansion velocity, where E = 3Q/(8πε 0 R) is the field at the cloud radius, Q = (4π/3)en 0+ R 3 0 is the full cloud charge, R = (3Q/(4πen + )) 1/3 is the cloud radius, e is the elementary charge, and µ + is the ion mobility.
Let us consider two cases, one for helium ions He + and another one for nitrogen ions N + 2 . Helium ions He + have the highest mobility, and ions of other types such as N + , N + 3 , N + 4 , NO + , O + 2 have mobilities close to the mobility of the chosen nitrogen ion, N + 2 . Therefore, these two cases can fully and qualitatively characterize our system. The ion mobility can be considered as constant 2 for the condition and takes the following values 2, 7 in our estimations: µ + = 10 cm 2 /V·s for He + and 2 cm 2 /V·s for N + 2 . The problem is formulated by solving a differential equation for the ionic cloud border motion 3 : Equation (1) is equivalent to the following equation: The solution to the equation (2) is as follows: The electric potential ϕ c at the center of the cloud will be ϕ c = en + R 2 /(2ε 0 ).
The decrease in time of the concentration n + for the two ion types and the potential ϕ c in the center of the ion cloud are shown in Figure S1. Current model does not include the effect of negative ions presence in the vicinity of the cloud. However, it is logical to consider that the main influence on the formation of the plasma jet regime is caused by the formation of a positive uncompensated charge cloud. The source of ions determined by He electron impact ionization and the ionization of nitrogen through the metastable states of helium atoms, which experience only slow diffusion.
A significant decrease in the potential and concentration occurs within a time of the order of tens of microseconds. In this case, various positive nitrogen ions will determine the final quasi-stationary distribution of the charge and potential in the system. This time value corresponds to the observed time at which the repetitive regime of the plasma jet formation is established and vice versa, as well as the relaxation time after the end of the voltage bunch.